complete embeddings of linear orderings and embeddings of lattice-ordered groups

ISRAEL JOURNAL OF MATHEMATICS, Vol. 56, No. 3, 1986

COMPLETE EMBEDDINGS OF LINEAR ORDERINGS AND EMBEDDINGS

OF LATTICE-ORDERED GROUPS

BY

M A N F R E D DROSTE Fachbereich 6 - - Mathematik, Universitiit GHS Essen, 4300 Essen 1, West Germany

ABSTRACT

An infinite linearly ordered set (S, < ) is called doubly homogeneous, if its automorphism group Aut(S, ~ ) acts 2-transitively on it. We study embeddings of linearly ordered sets into Dedekind-completions of doubly homogeneous chains which preserve all suprema and infima, and obtain necessary and sufficient conditions for the existence of such embeddings. As one of several consequences, for each lattice-ordered group G and each regular unconntable cardinal K _-> I G I there are 2 K non-isomorphic simple divisible lattice-ordered groups H of cardinality K all containing G as an /-subgroup.

§1. Introduction

An infinite linearly ordered set ("chain") (S, =< ) is called doubly homogeneous, if its automorphism group, i.e. the group of all order-preserving permuta- tions, A (S) = Aut((S, -_< )) acts 2-transitively on it. Chains (S, N ) of this type and their automorphism groups A (S) have been extensively studied. They have been

used e.g. for the construction of infinite simple torsion-free groups (Higman [8]),

or, in the theory of lattice-ordered groups (/-groups), in dealing with embeddings of arbitrary/-groups into simple divisible/-groups (Holland [9]). They also have

been used for the construction of certain partially ordered sets with transitive

automorphism groups ([2, 5]). Obviously, all linearly ordered fields are examples

for such chains. For a variety of further results see Glass [7].

As is well-known, any linearly ordered set can be embedded into a doubly

homogeneous chain (even into a linearly ordered field). However, these embed-

dings usually do not preserve arbitrary suprema or infima. Therefore here we

study embeddings of linear orderings or their Dedekind-completions into the

Received September 4, 1985 and in revised form April 20, 1986

315

316 M. DROSqE Isr. J. Math.

Dedekind-completion of doubly homogeneous chains which preserve all (exist-

ing) suprema and infima.

Before stating our results, let us introduce some notation. Let (C, -_< ), (D =< )

be two arbitrary chains. An embedding ~0 of ( C ~ ) into (D,=<) is called

complete, if ~o preserves all suprema and infima of subsets of C that happen to

exist in (C, =< ). We denote the Dedekind-completion of the chain (C,-<_) by

(t~, =<). A non-trivial closed interval of C is a set of the form [a, b] -- {c E C;

a = c =< b} where a, b E C with a < b. Recall that (C, -<) is dense if whenever

a, b E C with a < b, there is c E C with a < c < b. Now let (C, -<_ ) be dense and

unbounded. For each point a E C' we put

cof(a) = min{IA [; A C C, a E A, a = sup A }, the cofinality of a,

and

coi(a) = min{ [A I; A _C C', a E A, a = inf A }, the coinitiality of a.

If co f (a )= coi(a), this is called the coterminality of a. The chain (C, ~ ) (or

((~, =<)) has countable coterminality, if it contains a countable subset which is

unbounded both above and below in C. Using a result of Droste and Shelah [6],

we will show:

THEOREM 1. Let (C, <- ) be an arbitrary chain. The following are equivalent:

(1) No non-trivial closed interval of C is dense. (2) There exists a doubly homogeneous chain ( S, <-_ ) and a complete embedding

~p of ( C , < ) into (S,_-<) such that q~(C)C_S\S.

Moreover, if one of these conditions is saisfied and K >= I C I is any infinite cardinal,

then the chain (S, _<-) and the embedding ~p in condition (2) may be chosen such

that, in addition, S has cardinality K, each point s @ S has countable coterminality,

and whenever a,b E C with a < b and no c ~ C satisfies a < c < b, then the

interval (q~(a), qo(b )) in S has countable coterminality.

Note that the embedding result of Theorem 1 applies in particular to the class

of all scattered linear orderings, since any such chain satisfies the assumption (1)

of Theorem 1. A special case of Theorem 1 has already been used, in a natural

way, when constructing all normal subgroup lattices of the automorphism groups

A (S) (S a doubly homogeneous chain), cf. [6; §4]. Other special cases of this

result have been used in [3, 5].

Let again (C, ~ ) be an arbitrary chain. We will call a pair (A, B) of non-empty

disjoint subsets A, B of C a Dedekind-hole in C if C = A U B, a < b for all

a E A, b E B, and A has a greatest element iff B has a smallest element. The set

Vol. 56, 1986 COMPLETE EMBEDDINGS 317

of all Dedekind-holes of C is said to be dense in C if whenever a, b E C with

a < b , there exists a Dedekind-hole ( A , B ) of C with a E A , b E B . As a

consequence of Theorem 1 we will derive

COROLLARY 1. Let (C,<=) be an arbitrary chain. The following are equivalent:

(1) The set of Dedekind-holes of C is dense in C. (2) There exists a doubly homogeneous chain (S, <= ) and a complete embedding

q~ of (C, <) into (L <-<_ ) such that q~(C)C_S\S.

Moreover, if one of these conditions is satified, the chain ( S, <= ) and the embedding q~ in condition (2) may be chosen such that, in addition, S has cardinality I (71, each point s E S has countable coterminality, and whenever (A ,B) is a Dedekind-hole of C, then supq~(A)<infq~(B) in (S, <=) and the interval (sup q~(A),inf q~(B)) in S has countable coterminality.

Observe here that the class of all linear orderings with a dense set of

Dedekind-holes includes the chains satisfying condition (1) of Theorem 1, but

also dense chains like the rationals and the irrationals, more generally all sets C

for which C'\ C is dense in C. As a further consequence of Theorem 1 and a

result of Solovay [18], we will show:

THEOREM 2. For each regular uncountable cardinal K and all regular cardinals p., u <_ K, there exist up to isomorphism precisely 2 ~ doubly homogeneous chains (S, --< ) of cardinality K with pairwise non-isomorphic and non- antiisomorphic Dedekind-completions such that each point s E S has cofinality and coinitiality v.

We will also obtain an answer to a question of van Benthem [19] which arose in a study of tense logic and time.

For any chain (S,=<), A(S) is a lattice-ordered group (/-group) by the

pointwise ordering of functions (for f, g E A(S) , put f <= g iff f(s) <~ g(s) for all

s E S). If S is doubly homogeneous, then B(S), the /-subgroup (subgroup sublattice) of A (S) comprising all automorphisms of S with bounded support, is algebraically simple and divisible (Higman [8], Holland [9]). If G, H are/-groups

and ~b: G--~ H is a group-embedding which preserves the lattice operations (v

and A), then ~ is an /-embedding. Now let GC_A(S) , H C A ( T ) be l-

subgroups for some chains S, T, and let ~o: S--~ T be an embedding and

~b: G--~ H an/-embedding. Then (~, ¢) is called an / -embedding of (G, S) into (H, T), if q~ og = ~(g)o~0 for all g E G. We will show:

THEOREM 3. Let (C, <= ) be an arbitrary chain and K ~ I C I a regular uncount-

318 M. DROSTE Isr. J. Math.

able cardinal. Then there are 2 K doubly homogeneous chains ( S, <= ) of cardinality

K with pairwise non-isomorphic and non-antiisomorphic Dedekind-completions

such that ( A ( C ) , C) can be l-embedded into (A(S), S), or, in ]:act, even into

(B(S),S).

Note that this result contains, in particular, our starting-point that any chain

can be embedded into a doubly homogeneous chain as a special case again. For

many of the sets (S, ~ ) constructed here, the automorphism groups A (S) are

elementarily inequivalent as groups. Using Holland's theorem [9] that any

/-group G can be /-embedded into A ( C ) for some chain C and a result of

McClearly [15], we obtain as a consequence of Theorem 3:

COROLLARY 2. Let G be an arbitrary l-group and K >-_ ] G I a regular uncount-

able cardinal. Then there exist 2" simple divisible l-groups H of cardinality K

which are pairwise non-isomorphic as groups and all contain G as an l-subgroup.

Theorem 3 and Corollary 2 sharpen results of Holland [9] and Weinberg [20].

We will also consider embeddings of arbitrary /-groups into /-groups all of

whose group automorphisms are inner, and we show how to obtain similar

results for singular cardinals.

The proofs of Theorems 1, 2, 3 are contained in Sections 2, 3, 4, respectively.

For the basic background on our topic we refer the reader to Glass [7].

§2. Complete embeddings of linear orderings

In this section we prove Theorem 1 and Corollary 1 and derive a few related

results on complete embeddings. First let us show that embeddings of linear

orderings into (doubly) homogeneous chains which preserve all suprema (or all

infima) are rare. An infinite chain (S, _<-) is called homogeneous if A ( S ) acts

transitively on it.

PROPOSITION 2.1. Let (C, <_-) be a linear ordering for which there exists an

embedding of the ordinal tOl + 1 into C which preserves all suprema. Let (S, _-< ) be

any homogeneous chain. Then there is no embedding of (C, <= ) into (S, <-_ ) which

preserves all suprema.

PROOF. Assume there is an embedding q~ of (/)1 + 1 into (S,_- < ) which

preserves all suprema. Let a E to~ + 1 be a countable limit-ordinal and b the

maximal element of o)~ + 1. Then cof(~p(a))= No and cof(~(b))= ~t~ in S, thus

there is no automorphism of S mapping ~o(a) onto ~o(b). Hence (S,_-<) is not

homogeneous, and the result follows.


Now we turn to the proof of Theorem 1. We first need a few preparations. If

(C ,_ <-) is a chain, Z C_ C, and a,b ~ (~ with a < b, we let (a ,b)z ={z E Z ;

a < z < b}; also ( - w , a ) z ={z E Z ; z < a}. The intervals [a, b]z and (a,~)z

are defined similarly, and the index Z is omitted if this unambiguous.

DEFINITION 2.2 ([6]). Let r be a cardinal. A chain (C, _<-) is called a good

K-set, if the following conditions are satisfied:

(1) I CI = r, and (C, ~ ) is dense and unbounded.

(2) Each c ~ C has countable coterminality.

(3) Whenever x,y E C with x < y , there is a set A C[x,y]c,\C such that

t A I = r and each a E A has countable coterminality.

Obviously, any good l%-set is isomorphic to Q, the set of all rationals. Now we

deal with the existence problem of good r-sets for arbitrary cardinals K:

LEMMA 2.3 ([6; Lemma 4.2]). Let K be a cardinal. Then there exists a good K-set (C, <= ) of countable coterminality.

If (C,=<) is a chain and A , B C_ C, we write A < B (A =<B) to denote that

a < b (a_-<b) for all a ~ A , b E B . Also let x < A ( A < x , A ~ x etc.)

abbreviate { x } < A (A <{x}, A _<-{x}), respectively. Now let (S~,=<), ($2,_- < )

be two chains with (S~, _-< ) C_ ($2, ~ ), i.e. S~ _C $2 and the order of $2 extends the

order of $1. Then $1 is dense in $2 if whenever a, b E $2 with a < b, there is

x ~ S~ with a < x < b. Hence S, is dense iff $1 is dense in itself. Let a ~ S~. We

put

Ded(a, S1, S2)= {z E $2; {s E S~; s < a } < z <{s E S~; a < s}}.

For later purposes note that we have Ded(a, $1, $2)= Q iff a ~ $2 and for any

z E $2 with z <{s ~ S~;a < s} ({s E S1;s < a } < z) there is x E S~ with z =<x <

a (a < x _-< z), respectively. Now we give the

PROOF OF THEOREM 1. (2)---> (1): Let a, b E C with a < b. Since S is doubly

homogeneous, it is dense, and we can choose s @ S with q~(a)< s < q~(b). Let

A = { z @ C ; ~ ( z ) _ < - s } a n d B = { z ~ C ' ; s ~ q ~ ( z ) } , a n d p u t c = s u p A , d = i n f B

in (C,_-<). Then ¢,(c)=sup~p(A), tp(d)=infq~(B) in (S ,~ ) and q ~ ( c ) < s <

q~(d) by s E q~((~). Thus c < d, and no z E C' satisfies c < z < d. Hence c, d E C,

showing that the interval [a, b]c is not dense.

(1)--~(2): We show that we can find a doubly homogeneous chain (S, _-<)

satisfying also the additional properties stated in the theorem. So let K _--> ] C] be

any infinite cardinal. We may assume that C has no greatest and no smallest

element (otherwise "enlarge" C correspondingly). First let us assume that K is

regular.


Let P be the set of all pairs (a J b) such that a,b ~ C, a < b, and no c ~ C

satisfies a < c < b. For each pair (a [ b) ~ P choose by Lemma 2.3 a good K-set

L<~I~ with countable coterminality, and let

So= C_J L~alb~. (arb)~P

Next we define a linear order on (~t] So in the natural way such that it extends

the given orders of C and of each L~,lb~ and a < L(alb) < b in (Ct] So, ---- ) for each

(a I b) E P. Note that whenever c, d E C with c < d, by assumption (1) there are

a, b E C such that c _-< a < b =< d and (a I b) E P. This shows first P ~ ~ , hence

ISo] = K and (So, ----) is a good K-set. Secondly, we also obtain that whenever

c, d E (~ with c < d, there is s E So with c < s < d. Thus So is dense in So t] C',

and hence we may regard (7 as a subset of So\ So. Thirdly, C is unbounded above

and below in So.

Now if K = 1~0, we put S = So and let ¢ be the identity mapping. Since then S

is isomorphic to Q, A (S) is doubly homogeneous. As q~ is complete (as indicated

in detail below), the result follows. Therefore let now K be regular and

uncountable. We follow the construction in the proof of Theorem 2.11, parts

(I)-(III), of Droste and Shelah [6] almost literally, with To = C, A = K, fL = S~.

Hence there are chains (S,, <_- ) (i E K) such that IS, I = K and (So, =< ) C_ (S~, ~ ) C

(Sj,_- < ) for all i, j U K with i< j , Ded(a, So, S i )=Q for each a E ( ~ (cf. [6;

conditions (4.7) (I), (III)]), and if (S, _-< ) is defined in the natural way such that S = I..Ji~K S~ and the order of S extends the order of each S~ (i E K), then (S, =<)

is a doubly homogeneous chain and each point s E S has countable coterminal-

ity. It follows that I SI = K and also Ded(a, So, S ) = Q for each a G C'. In particular, this shows C f) S = ~ and if a ~ tO-, A C S~, and a = supA (a =

i n f A ) in SoQI C, then also a = s u p A (a = i n f A ) in S U C. Hence (~ is also a

subset of S, thus C C_ S \ S, and we may define ~ to be the identity mapping.

Clearly q~ preserves all suprema and infima of subsets of t~, since if e.g.

a = s u p A in C for some a U t~ and A C C_ C, then also a =sup B in So for

B = {s ~ So; s < a} by C C S,,. Hence also a = sup B in S; now the fact that A is

unbounded above in A t_J B implies that a = sup A in S. Also, whenever

(a I b) E P, the set Lt, lb) is unbounded above and below in {s ~ S: a < s < b}.

Since L(orb) has countable coterminality by construction, it follows that (a, b)s

also has countable coterminality. Hence we have shown the result in this case.

If K is a singular cardinal, we proceed in almost precisely the same way. We

only have to take into [6; Definition 4.9] the additional requirement that if A is a

singular cardinal, then also IJump(O~,iqk)l=<~ whenever i < k < j (in the

VOI. 56, 1986 COMPLETE EMBEDDINGS 321

notation present there). Then [6; Lemma 4.10] holds also for singular cardinals A. Observe that in [6; Lemma 4.8] we have also

I Jump(l~,, ~~3)I ~ IJump(~,, f~2) l + t Jump(l)2,113) I,

as shown there in the proof. Now our proof goes through as before.

The following result, an immediate consequence of the argument just given,

was also noted by S. Shelah.

COROLLARY 2.4. Theorem 2.11 of Droste and Shelah [6] holds also for singular cardinals A.

We just note that now the use of Theorem 1 allows us to considerably simplify

the argument for Theorem 2.11 of Droste and Shelah [6] (note that the set To

constructed in the proof of [6; Theorem 2.11] satisfies assumption (1) of

Theorem 1, apply Theorem 1 to obtain the doubly homogeneous chain (1~, =< ),

and continue with part (IV) of the proof of Theorem 2.11).

The existence result of Theorem 1 will be sharpened in Theorem 3.2. We note

that the chain S constructed in the above proof has some additional interesting

point character properties. For this, we need some more notation. Let (D, _-< ) be

a Dedekind-complete chain, not necessarily dense. Let a E D. If a has no

immediate predecessor (successor), we define the cofinality (coinitiality) of a as

before; otherwise we put c o f ( a ) = No (co l (a )= No), respectively. Then the pair

(cof(a),coi(a)) is called the character of a. Let Char(D) denote the set of all

characters of elements a E D. We wish to determine Char(S). It is easy to check that the good K-set L constructed in [6; Lemma 4.2] (and used in the above proof) satisfies

Cha r ( i ) = {(t x, No); No--< p, = K,/x regular}.

Clearly Char(/~) LI Char(C') _C Char(S). We may even obtain equality here in the following way. Whenever j < K is a countable limit-ordinal, I C K with j = sup I, and a,, b~ E &+~\S, with a, < a,+l < b,+l < b~ for each i E I, then in Sj = I,..J~<jS, we have

sup{a, ; i E I} = inf{b, ; i E I} =: x E ~ \ Sj

and x has coterminality No in Sj. In order to get Ded(x, S j , S ) = O and hence

cof(x) = co i (x)= No in S, we let Tj, the set of all "forbidden points" of Sj\Sj,

contain (_J,<j T~ and all elements x ~ Si\Sj of the form described above (cf. [6;

pp. 251, 255]). Continuing with the construction of S as before, we obtain:


COROLLARY 2.5. Let ( C, <= ) be a chain in which no non-trivial closed interval

is dense, and let K >=ICI be any infinite cardinal. Then there exists a doubly

homogeneous chain (S, _-< ) of cardinality K satisfying the assertion of Theorem 1,

and, in addition,

Char(S) = {(/z, no);/z ~ K,/z regular} U Char((~).

Next we use Theorem 1 to give the

PROOF OF COROLLARY 1. (1)--->(2): We show that we can find a doubly

homogeneous chain (S, =< ) satisfying also the additional properties stated in the

corollary. Let H be the set of all Dedekind-holes (A, B) of C for which A has no

greatest and thus B no smallest element. For each (A, B ) E H, choose two

elements aA, RE,. Let

D=C0 0 IRA,a°}, (A,B)EH

and define a linear order -_< on D in the natural way such that it extends the

order of C and A <aA < a ~ < B in (D,<=) for each ( A , B ) E H . Then I D t =

] (~[, D is Dedekind-complete, the identity mapping which embeds (C, <_- ) into

(D, =<) is complete, and the assumption (1) shows that no non-trivial closed

interval of D is dense. Hence we may apply Theorem 1 on (D,=<) which

immediately implies the result.

(2)--~(1): Let a, b E C with a < b . Since S is dense, there is s E S with

~o(a)<s<q~(b). Let A = { c ~ C ; q ~ ( c ) < s } and B = { c C C ; s < ¢ ( c ) } . Then

a ~ A , b C B, C = A 0 B as s ~ ¢(C) , and A < B. It remains to show that A

has a greatest element iff B has a smallest element. So assume without loss of

generality that A has a greatest element c E A. If B has no smallest element, we

would obtain c = inf B in (C, _-< ) and thus ~0(c) = inf q~(B) in (S, =< ) in contradic-

tion to ~o(c) < s < q~(B). Hence B has a smallest element, and the result follows.

In view of Theorem 1 we now show that not every chain can be completely

embedded into the Dedekind-completion of a doubly homogeneous chain.

PROPOSITION 2.6. Let ( C, <= ) be a dense unbounded chain such that (C, _<-) is

not the Dedekind-completion of a doubly homogeneous chain. Let (S, _-< ) be any

doubly homogeneous chain. Then there is no complete embedding q~ of (C, _-< ) into

PROOF. Suppose such a complete embedding q~ exists. Then, as shown in the

proof of Theorem 1, ~0(C') is a convex subset of S, i.e. whenever a, b @ C' and


s E S with q~(a)< s < ~0(b), then s E q~(i). Hence q~(i) is the Dedekind-

completion of ~ ( i ) D S, and the chain q~(i)N S is doubly homogeneous, a

contradiction.

However, Propositions 2.1 and 2.6 are contrasted by the following conse-

quence of Theorem 1.

COROLLARY 2.7. Let (C ,= < ) be an arbitrary chain and K >=I CI any infinite cardinal. Then there exists a doubly homogeneous chain (S, =< ) of cardinality K

and an embedding q~ of ( i , <= ) into (S, <= ) such that q~ preserves all suprema of subsets of i .

PROOF. Let D = C x {1,2} be ordered lexicographically, i.e. (c, i) < (c', j) iff

either c < c' or c = c' and i < j (c ,c 'E C,i,j E {1,2}). Then /3 = DO ( i f \C) in

the natural way such that (c, i) < 6 iff c < 6 ((c, i) ~ D, ~ ~ if \C). Clearly the

mapping lr: i - -~/3, defined by 1r(c) = (c, 1) if c ~ C, and It(c) = c if c E i \ c ,

is an embedding and preserves all suprema of subsets of i . Since J D I --< K and no

non-trivial closed interval of (D, =< ) is dense, now Theorem 1 implies the result.

§3. Constructions of non-isomorphic doubly homogeneous chains

In this section we use Theorem 1 and a result of Solovay [18] to construct

"many" doubly homogeneous chains of a prescribed cardinality with non-

isomorphic Dedekind-completions and with several additional properties. In

particular, we will prove Theorem 2.

First let us sharpen the main result of Theorem 1. As usual, we identify cardinals with the least ordinals of their cardinality. Let (S, =< ) be a chain with no greatest element. Let us call a subset A C_ S a club in S, if it is closed and

unbounded above in S, and stationary in S, if it intersects each club of S. The

cardinal

cof(S) = min{IA [; A C_ S is unbounded above in S}

is called the cofinality of S. Dually, if S is unbounded below, the cardinal

col(S) = min{IA I; A C_ S is unbounded below in S}

is called the coinitiality of S. If cof(S) = col(S), this is called the coterminality of

S. The proof of the following lemma is straightforward.

LEMMA 3.1 (cf. [1; 3.16(C)]). Let S be an unbounded chain with uncountable cofinality and A, B C_ S two clubs in S. Then A (3 B is also a club in S. I f

324 M. DROS'IE Isr. J. Math.

~b: A ~ B is an isomorphism, there exists a club C C A fq B on which ~b acts like the identity.

Now we show:

THEOREM 3.2. Let (C, -_<) be an arbitrary chain in which no non-trivial closed

interval is dense, h any regular uncountable cardinal, and K >= )t + I C I. Then there exist (at least) 2 ~ doubly homogeneous chains (S, <-) of cardinality K and

cofinality h with pairwise non-isomorphic and non-antiisomorphic Dedekind-

completions such that for each of these chains (S, <-_ ), there exists a complete

embedding q~ of (C, -< ) into (S, _-< ) with the following properties: (i) ~(C') C_ S \ S ;

(ii) whenever a, b E C with a < b and no c E C satisfies a < c <b, then the interval (~(a ), tp(b )) in S has countable coterminality ;

(iii) each point s E S has countable coterminality.

PROOF. We may assume that (C,-_ < ) contains a copy of the ordinal K;

otherwise enlarge (C, _-< ) correspondingly. Let (7-,-< ) be a well-ordered chain

isomorphic to h. For each element a E T, choose a chain (L,,_-<) which is

antiisomorphic to tot. For each subset Z C_ T let

a C Z

Next we define a linear order =< on Cz such that it extends the orders of C, T,

and of each L~ (a E Z), C < (TU U ~ z L ~ ) , and a < Lo < min{t E T; a < t} for

each a E Z. Then I Cz [ = r, and clearly no interval of Cz is dense. Hence by

Theorem 1 there is a doubly homogeneous chain (Sz, <= ) of cardinality r such

that (without loss of generality) Cz C_ Sz\ Sz, the identity mapping from Cz into

Sz is complete, each point s E Sz has countable coterminality, and whenever

a, b E Cz with a < b and no c E Cz satisfies a < c < b, then the interval (a, b) in

Sz has countable coterminality. Note that hence in Sz we have coi(a) = I~ for

each a E Z , and co i (a )= l% for each a E T \ Z . We may assume that Cz is

unbounded above in Sz; otherwise let s = sup Cz ~ Sz and consider {x E Sz ; x < s} instead of Sz. In particular, Sz has cofinality h. We can also assume that

Sz has countable coinitiality; otherwise choose s E Sz with s < Cz and coi(s) =

No and consider {x E Sz ;s < x} instead of Sz. If follows that the chains Sv, Sz (Y, Z _C T) are never antiisomorphic. Note that since Cz contains a copy of r,

this procedure does not change the cardinality of Sz. In fact, since this copy of x

is bounded in Sz, a copy of r is contained in each non-trivial interval of Sz.


By a well-known result of Solovay [18], we can split T = Ui~ ,A , such that

each A, is a stationary subset of T (i E X). For each A C )t, let A* = I,.J~ea A~ C

T. We claim that the chains SA. (A C A ) satisfy the requirements of the theorem.

For this, it only remains to show that whenever A, B C ~ with A / B, then Sa-

and S~. are not isomorphic.

Indeed, suppose O: S,~(--* S~. were an isomorphism. Since T and qJ(T) are

subsets of SB. which are closed and unbounded above in S~[, by Lemma 3.1

there is a club D C_ T fq O(T) in T on which ~b acts like the identity. Since A / B,

we may assume that there is some i C A 1B. Then A, C A* C_ T, and since A, is

stationary in T, there exists a E A, (q D C T C SA.. Consequently, a has coini-

tiality N~ in SA. and, by a = q~(a), also in S~.. But A~ is disjoint to B*, hence

a E T \ B* C SB- must have countable coinitiality in SB., which establishes our

contradiction.

If (T, _-__ ) is a chain and x E T, we call (x,0c)r = {t E T; x < t} a .final segment

of (T, =<). As a consequence of the proof of Theorem 3.2 we note that the

constructed doubly homogeneous chains (S,_-<) have the additional property

that even the final segments of their Dedekind-completions are pairwise

non-isomorphic and that each non-trivial interval of (S, _-< ) contains a copy of K.

The following consequence of Theorem 3.2 will be applied in [5] for the

construction of certain infinite partially ordered sets with transitive automor-

phism groups.

COROLLARY 3.3. Let A be a regular uncountable cardinal and r >-_ A. Then

there are (at least) 2 ~ doubly homogeneous chains (S, <=) o[ cardinality K and

cofinality A with pairwise non-isomorphic and non-antiisomorphic Dedekind-

completions such that for all pairs of regular cardinals Ix, v <-_ K, there exists a point

s @ S \ S with cofinality Ix and coinitiality v. Moreover, these chains (S, _-< ) can be

chosen such that even the .final segments of their Dedekind-completions are

pairwise non-isomorphic, and such that each non-trivial interval of each of the

chains (S, =<) contains a copy of the ordinal r.

PROOF. Let P be the set of all pairs (/z, v) such that /z, v =< r are regular

cardinals. Well-order P. For each p = (p., v) @ P, choose a chain Ap isomorphic

to/x, a chain Bp antiisomorphic to v, and a point Zp not belonging to Ap or B v.

Let Cp = Ap t] {zp} U Bp and define a linear order _-< on Cp such that it extends

the orders of Ap and Bp and Ap < {Zv} < Bp. L.et (C, =< ) be the ordinal sum of the

linear orders (C,, < ) (p E P); hence C = Up~pCp and p,q @ P, p < q imply

Cp < Cq in (C, _--- ). Now apply Theorem 3.2 to obtain 2 ~ doubly homogeneous


chains (S, -<_ ) of cardinality K and cofinality A with pairwise non-isomorphic and

non-antiisomorphic Dedekind-completions, and complete embeddings ~ of

(C,-<_ ) into ( S , - ) . Then for each p = (/x, v ) E P, the element q~(zp)E S \ S has

cofinality/z and coinitiality u. The final statement of the corollary is immediate

by the remark following Theorem 3.2.

In applications of Theorem 3.2 and Corollary 3.3 (as, for instance, for our

proof of Theorem 2), of course the most important cases are the ones where

either K = A or else K is a singular cardinal. Now we turn to the proof of

Theorem 2. Let (S, =<) be any chain. Each automorphism f of (S, =< ) extends

uniquely to an automorphism of (S, _-< ) which we will also denote by f. If z @ S,

the set {f(z); f ~ A ( S ) } is called the A(S)-orbit of z in S.

LEMMA 3.4 (McCleary [14; p. 418]). Let (S ,N) be a doubly homogeneous

chain, z ~ S, and U the A ( S )-orbit of z in S. Then U is a doubly homogeneous

chain, U is dense in S, and

col(x) = cof(z), coi(x) = coi(z) for all x E U.

We will also need the following L6wenheim-Skolem-type result for doubly

homogeneous chains.

LEMMA 3.5. Let C be a doubly homogeneous chain, n a cardinal, and A C C such that [ A I <= K <= I C I. Then there exists a doubly homogeneous chain B with A C_BC_C and IBI=K.

PROOF. By assumption, there exists a function f: C5--~ C such that for all

a, b, c, d ~ C with a < b and c < d, the function f(a, b, c, d,. ) maps the interval

[a,b] isomorphically onto [c,d]. By the downward L6wenheim-Skolem

theorem the model ~ =(C, < ,f) has an elementary submodel ~3 < ~ whose

domain B satisfies A C B _C C and I B I = K. The result follows.

Next we give the

PROOF OF THEOREM 2. Let K be any regular uncountable cardinal. Clearly

there are at most 2 K non-isomorphic chains of cardinality K. To prove the

existence part, by Corollary 3.3 we can choose 2 K doubly homogeneous chains

(S~, _-< ) (i < 2") of cardinality and cofinality K with pairwise non-isomorphic and

non-antiisomorphic Dedekind-completions such that there exists a point s, E

S,\ S, with cofinality/z and coinitiality u, and each non-trivial interval of (S,, _--- )

contains a copy of K. NOW, for each i < 2 K, choose such a point s~ and let Ui be

the A(S~)-orbit of si in S~. By Lemma 3.4, U~ is doubly homogeneous, U~ is

VOI. 56, 1986 COMPLETE EMBEDDINGS 327

dense in S~, and cof(x) =/~, col(x) = v for all x E U~. For all a, b E S~ with a < b

choose an element c E U~ with a < c < b, and let Di be the set containing precisely all these elements c = c(a,b). Thus D, C U,, and D~ is dense in S,

Therefore I D~ ] = K, since K can be embedded into S~. By Lemma 3.5, there is a

doubly homogeneous chain Z of cardinality K such that D~ C_ Z C_ U~. Thus Z is

also dense in S~, showing that Z = S~ and col(x) =/~, coi(x) = v for all x ~ %.

Consequently, the chains T~ (i < 2 ~) satisfy the requirements of the theorem.

For algebraic properties of the automorphism groups A (S) constructed here,

see section 4. We just note that if K is a singular cardinal, by the same argument

there are at least 2 ÈK = X~(~2 ~ doubly homogeneous chains (S, _<- ) of cardinality

K satisfying the requirements of Theorem 2. Moreover, we can assume that each

non-trivial interval of each of the chains (S, =< ) and, consequently, also of (S, =< ),

contains a copy of the ordinal K. As a consequence of this remark, we obtain the

following positive answer to a question of van Benthem [19; p. 45]. Under the

generalized continuum hypotheses, for regular cardinals K this result has already

been established by McCleary [14; Theorem 15]. For ~ = N~, it is contained in

Burgess [22].

COROLLARY 3.6. For each uncountable cardinal K there exists an infinite chain

(S, _-< ) of cardinality K such that each open interval (a, b) orS (a, b E S, a < b) is

isomorphic to (S, = ), but (S, _-< ) is not antiisomorphic to itself.

PROOF. By Theorem 2 and the above remarks, there exists a doubly

homogeneous chain (S',_-<) of cardinality K such that each point s E S' has

cofinality ~tt and coinitiality N0, and that each non-trivial interval of S' contains a

copy of K. Choose a, b E S' with a < b and put S = (a, b). The result follows.

§4. Embedding results for lattice-ordered groups

In this section we wish to prove Theorem 3 and derive from it universal

embedding results like Corollary 2. We will also consider embeddings of

arbitrary/-groups into/-groups all of whose group automorphisms are inner. As

in the previous section, we will frequently obtain collections of doubly

homogeneous chains (S~, < ) with pairwise non-isomorphic and non-

antiisomorphic Dedekind-completions. We just mention here that then an

obvious consequence for the corresponding automorphism groups A(S~) is

immediate by the following result. Let e always denote the identity element in a

given group.

328 M. DROSTE lsr. J. Math.

PROPOSITION 4.1 (Holland [10], J ambu-Gi raude t [11], McCleary [15],

Rabinovich [16], cf. [7; §2]). Let S, T be two doubly homogeneous chains whose

Dedekind-completions S, T are neither isomorphic nor antiisomorphic. Then the

automorphism groups A (S) and A (T) are non-isomorphic both as groups and as

lattices with e.

Now we turn to the proof of T h e o r e m 3. We will need the following lemma.

LEMMA 4.2. Let (S,<=) be a doubly homogeneous chain and a~,b~ E S \ S,

s~ E S such that a~ < s~ < b~ and coi(a~) = cof(b~) = No for i = 1,2. Then there is an

isomorphism lr from (al, b~)s onto (a2, bz)s mapping sl onto s2.

PROOF. For i = 1,2, choose a countable subset G = {cu ; j E Z} C_ (ai, b~)s

which is unbounded above and below in (a~, b~)s such that cu < cu+~ for each

j E Z and c~.o = si. For each j ~ Z, choose an isomorphism 1r / f rom [c~.j, cLj+~]s

on to [c2,j, c2.j+l]s, and put ~r = Uj~z 7ri to obtain the result.

Recall that we identify each au tomorph ism of a chain (S, _-< ) with its (unique)

extension to (S, =< ). Now assume that S and S \ S are dense in (S, =< ). Then each

au tomorphism of (S, _-< ) maps S \ S onto itself; hence we can identify A ( S ) with

A ( S \ S ) , in a natural way. This is done in the following result. Its proof uses

ideas a l ready conta ined in Hol land [9; proof of T h e o r e m 4].

THEOREM 4.3. Let (C, <=) be a chain, (S, <=) a doubly homogeneous chain,

and q~: (C, =< )---~ (S, <= ) a complete embedding such that q~ (C) C S I S. Assume

that whenever a, b E C satisfy a < b and there is no c E C with a < c < b, then the

interval (q~(a),q~(b)) in S has countable coterminality. Then there exists an

l-embedding ~b of A ( C ) into A ( S ) such that (~, ~) l-embeds ( A ( C ) , C) into

( A ( S ) , S \ S ) and ~b maps B ( C ) into B(S) .

PROOF. L e t P b e t h e s e t o f a l l p a i r s ( a l b ) E C x C s u c h t h a t a < b a n d t h e r e

is no c E C with a < c < b. For each pair (a I b ) E P let Dtoib~ = (q~(a), q~(b))s.

By assumption and L e m m a 4.1, the sets Dt~lb ~ ((a I b ) E P ) are pairwise

isomorphic. We now define a family of isomorphisms 7r,, : Dq ~ Dr (q, r E P) as

follows. Fix p E P and let ~rp, p be the identity mapping on Dp. For each q ~ P

with q ~ p, choose an isomorphism 7rp,, from Dp onto Dq, and let 7r, p = fr,~q. For

any q, r E P, let 7r,,, = 7rp., o ~q.~. Then 7r,, = 7r,,, o 7r,, for any q, r, t E/9.

Next let s ~ S such that q~(a) < s < q~(b) for some a, b ~ C. We claim that

s ~ D~ for some q E P. Indeed, let A = {z ~ C'; ¢p(z) =<_ s} and B = {z ~ C; s =<

¢p(z)}, and put c = sup A, d = inf B in ((~, =< ). As in the proof of T h e o r e m 1 we

obtain (c I d) ~ P and thus s ~ D(¢le ~.


Now we define a mapping tO: A(C)---~A(S). Let f ~ A ( C ) . We define

f* E A (S) as follows. For each q = (a ! b) E P, observe that r = (f(a) If(b)) c P, and put f* Ioq = 7rq.,. For each s E S with s < g,(C) or q~(C)< s let f * ( s ) = s.

Then f* is defined on all of S as shown above, and f * E A ( S ) . Now put

tO(f) = f*. Clearly tO is a group homomorphism, preserves the lattice operations, and

maps B(C) into B(S). We show that tO is injective, Let f E A(C) such that f* is the identity on S. Then f (a) = a, f(b) = b for each pair (a [ b) ~ P. By Theorem

1, for any c, d C C with c < d there is a p a i r ( a [ b ) E P with c_-<a<b_-<d.

Hence f is the identity on C. Consequently, tO is an /-embedding.

To prove that (to, q~) is an /-embedding, let f ~ A ( C ) . We claim that

q~ of = f* o q~ (where f* is identified with its extension to an automorphism of S).

Clearly, if c E{a,b} for some (a ] b ) E P, we have (q~of)(c) = (f*oq~)(e). Since

again no non-trivial closed interval of C is dense, and q~ is a complete

embedding, we obtain this identity for all c ~ C'. The result follows.

Note that as one of several immediate consequences of Theorem 1 (or

Theorem 3.2) and Theorem 4.3 we have

COROLLARY 4.4. Let (C, <= ) be an arbitrary chain and K >= I CI an uncountable cardinal. Then there exists a doubly homogeneous chain ( S, <= ) of cardinality K and an 1-embedding (tO, q~) of (A(C), C) into (A(S), S) such that q~ preserves all suprema of subsets of C and tO maps B(C) into B(S). If no non-trivial closed interval of C is dense, here q~ can be chosen to be complete.

PROOF. The final statement is immediate as indicated above. This implies the

first assertion as follows. Let (C,=<) be an arbitrary chain, and put D =

C x {1,2}, ordered lexicographically. Then, as shown in the proof of Corollary

2.7, there exists a natural / -embedding (tO, q~) of (A(C), C) into (A(D), 19) such

that ~o preserves all suprema of subsets of C. Now the result follows from the

final statement of the corollary.

Next we apply Theorem 3.2 and the proof of Theorem 4.3 to obtain the

PROOF OF THEOREM 3. First let (C ÷, ~ ) be the chain C together with a new

greatest and a new smallest element. Canonically, A ( C ) - - B ( C + ) = A ( C + ) .

Next let D = C + x {t,2} be ordered lexicographically. We identify C + with the

set C+x{1} in the natural way. Clearly ]D]_- < K and no non-trivial closed

interval of D is dense. By Theorem 3.2, there are 2 ~ doubly homogeneous chains

(S,_-<) of cardinality K with pairwise non-isomorphic and non-antiisomorphic

330 M. DROSTE lsr. J. Math.

Dedekind-completions such that for each of these chains (S, _<-), there exists a

complete embedding q~ of (/3, _-< ) into (S, -<_ ) and the triple D, S, q~ satisfies the

assumptions of Theorem 4.3.

We now use the same notation as in the proof of Theorem 4.3 (with C

replaced by D) and proceed similarly. For each pair q = (a t b) E P choose an

element sq E D,, and, by Lemma 4.2, choose the isomorphism %., such that it

maps Sp onto sq. Then, for any q, r E P, 7rq,, maps sq onto s,.

Now we construct the /-embedding 0: A(D)---~A(S) as before. Note that

each automorphism of C extends uniquely to an automorphism of D, and hence

A(C) is an /-subgroup of A(D). Next define an embedding q~* of (C, =< ) into

(S,= < ) as follows. If c E C , then q=((c, 1)l(c,2))EP; put q~*(c)=Sq. It is

straightforward to check that then (0, q~*) is an /-embedding of (A(C), C) into

(B(S), s).

Next we obtain the following embedding result for arbitrary /-groups.

COROLLARY 4.5. Let G be an arbitrary l-group and K >=IGI a regular uncountable cardinal. Then there exist 2" doubly homogeneous chains (S, <=) of cardinality K with pairwise non-isomorphic and non-antiisomorphic Dedekind- completions such that G can be l-embedded into B(S).

PROOF. By Holland's theorem [9], there exists a chain (C, =< ) with cardinality

ICI =<IGI such that G can be /-embedded into A(C). Now Theorem 3 implies

the result.

Let (S, =< ) be an infinite chain and G C_ A (S) a subgroup. Then (G, S) is called

a triply transitive ordered permutation group, if whenever A, B C S each have

three elements, there exists g @ G with g(A) = B. An element g E G is called

strictly positive, if g > e in A(S). For the proof of Corollary 2 we will need the

following special case of a result of McClearly [15]. It generalizes the group-

theoretical part of Proposition 4.1.

PROPOSITION 4.6 (McClearly [15; Main Theorem 4]). Let (G, S) and (H, T) be two triply transitive ordered permutation groups such that G and H each contain

strictly positive elements of bounded support. If G and H are isomorphic (as groups), then ( S, <= ) and ( 7", <= ) are either isomorphic or antiisomorphic.

Using this result and Corollary 4.5, we now give the

PROOF OF COROLLARY 2. By Corollary 4.5, there are 2 ~ doubly homogeneous

chains ( S . = < ) of cardinality K with pairwise non-isomorphic and non-


antiisomorphic Dedekind-completions such that, without loss of generality, G is

an /-subgroup of each B(Si) (i < T ) . For each i < T , we proceed as follows. Choose a strictly positive element zi ~ B(S,) and for each pair (A, B) of three- element-subsets A, B C_ S~ an automorphism [~ (A, B) C B(S~) which maps A

onto B. Let G~ be the /-subgroup of B(S~) generated by G, z~, and the

automorphisms ~ (A, B). Then [G~[ = K. Since B(S~) is simple and divisible, by

an easy induction we now obtain a simple divisible/-subgroup//~ of B(S~) which

contains Gi and has cardinality K. By Proposition 4.6, the groups Hi (i < 2 ~) are

pairwise non-isomorphic. The result follows.

Again, if K =>IGI is a singular cardinal, there are at least 2 <~ pairwise

non-isomorphic /-groups H of cardinality K satisfying the requirements of Corollary 2.

Next we consider embeddings of arbitrary/-groups into/-groups all of whose

group automorphisms are inner. It has been shown that if G is an /-group and

K => I G I a regular uncountable cardinal, then there exists a doubly homogeneous chain (S,<=) of cardinality IS[ = 2,<~2" such that G can be /-embedded into

A (S) and each automorphism of A (S) is inner (McClearly [14, 15], Weinberg

[21]). Here we wish to improve this result with regard to the cardinality of the

chain S, and to show that there are many non-isomorphic chains S (of a given

cardinality) with the above properties. We will need the following result

characterizing group-automorphisms and l-group-automorphisms ( = l-

automorphisms) of A (S) (S doubly homogeneous).

PROPOSITION 4.7 (Lloyd [12], Holland [10], McClearly [13], cf. [7; Theorem 2.4.1]). Let (S, =<) be a doubly homogeneous chain.

(a) If for no z @ S\S, the A(S)-orbit of z in S is isomorphic to S, then every l-automorphism of A(S) is inner.

(b) If (S, --- ) is not antiisomorphic to itself, then every automorphism of A (S) is an l-automorphism.

The following result is well-known (cf., e.g., [7; Lemma 1.10.10]).

LEMMA 4.8. Let ( S , ~ ) be a doubly homogeneous chain. If x, y E S \ S both have countable coterminality, they belong to the same A (S)-orbit in S.

Now we can show:

THEOREM 4.9. Let G be an arbitrary l-group and K >~ I GI any uncountable

cardinal. Then there exists a doubly homogeneous chain ( S, <= ) of cardinality K


such that G can be l-embedded into A(S ) and every automorphism of A (S ) is inner.

PROOF. By Holland's theorem [9], we can assume that G C_ A(CO for some

chain C1 with I C~ I -~ I G I. Let D = C~ × {1, 2} be ordered lexicographically. Next

let T be a copy of oJ1, the first uncountable ordinal, and let (C, ~ ) be the ordinal

sum of (D, _-< ) and (T, ~ ), i.e. C -- DU T and D < T in (C, ~ ). By Theorem 3.2

and the remark following it, there exists a doubly homogeneous chain (S, ~ ) of

cardinality K and a complete embedding ~ of (C, ~ ) into (S, ~ ) such that each

non-trivial interval of S contains a copy of K, and conditions (i)-(iii) of Theorem

3.2 are satisfied. We may assume that ~(T) is unbounded above in S. Indeed,

otherwise let s = s u p s ( C ) and consider {x C S ;x < s} instead of S. Conse-

quently, S has cofinality ~ . Since each non-trivial interval of S contains a copy

of K, this procedure does not change the cardinality of S. Similarly, we may

assume that S has countable coinitiality. Hence S is not antiisomorphic to itself.

Clearly, A(CO can be /-embedded into A(D) , A ( D ) into A(C), and, by

Theorem 4.3, A(C) into A(S) . Hence it only remains to show that each

automorphism of A (S) is inner. By Proposition 4.7, it suffices to show that there

is no A (S)-orbit Z in S \ S which is isomorphic to S. Suppose there were such an

A(S)-orbit Z in S \ S and an isomorphism ~b" Z-->S. Since each s ~ S has

countable coterminality, the same applies to all z E Z. By Lemma 4.8, Z

contains precisely all points x ~ S \ S which have countable coterminality. Let ~b also denote the extension of ~b to an automorphism of S. The set A = ~(T), and

hence also B = ~(A), is closed and unbounded above in S. By Lemma 3.1, there

exists a club E _C A N B on which ~b acts like the identity. Thus E N Z = ~ .

However, A is isomorphic to to~, and our assumption on ~ implies that each

element of A (except maybe the first) has countable coterminality, showing

E C_ A _C Z U {rain A }, a contradiction. The result follows.

We remark that it is not difficult to combine the ideas of the proofs of

Theorems 3.2 and 4.9 in order to sharpen the existence result of Theorem 4.9.

We only have to split the (stationary) subset of all countable limit-elements of T

into two disjoint stationary subsets T~ and T2, and perform by Solovay's result

the splitting-argument of the proof of Theorem 3.2 only on the set T2 (instead of

T). Then each element of ~ (T 0 has countable coterminality in S, and in the

above proof of Theorem 4.9 we obtain the final contradiction by E N ~p(T~)~ Q.

Hence we have shown:

COROLLARY 4.10. Let G be an arbitrary l-group and K >= I GI an uncountable

cardinal. If K is regular, there exist 2 ~ doubly homogeneous chains (S, <-) of


cardinality K with pairwise non-isomorphic and non-antiisomorphic Dedekind-

completions such that G can be 1-embedded into A ( S ) and every automorphism of

A (S) is inner. I f r is singular, there are at least 2 <~ = Y~ <K 2 ~ such chains (S, <-_ ).

As pointed out earlier, all these automorphism groups A ( S ) are non-

isomorphic both as groups and as lattices with e. Indeed, many of the doubly

homogeneous chains (S, =<) can be constructed in a way such that the au-

tomorphism groups A ( S ) are even elementarily inequivalent as groups (or,

equivalently, as lattices with e; cf. [7, 11]). We put this remark into a more

general context. For any group G, let (g) denote the smallest normal subgroup of

G containing g @ G; we put NI(G) = {(g); g E G}. In [1, 3, 4, 6-9] the structure

of the normal subgroup lattice of the groups A (S) (S a doubly homogeneous

chain) was studied; it is determined uniquely by the structure of its partially

ordered subset (N~(A(S)) , C) . The following result is essentially contained in

Ball and Droste [1].

PROPOSITION 4.11. Let S, T be two doubly homogeneous chains. I[ A (S) and

A ( T ) are elementarily equivalent as groups, then ( N ~ ( A ( S ) ) , C ) and

(NI (A(T) ) , C ) are elementarily equivalent as partially ordered sets.

PROOF. As shown in [1; Theorem 3.12], there is a single formula q~(x, y) in

the first order language of predicate calculus for group theory such that

whenever f, g E A ( S ) , then q~[f, g] holds in A (S) if[ (f)C_ (g). This implies the

result.

Now the results of [3; (7.6)-(7.8)] show that for many of the doubly

homogeneous chains (S,-<) constructed in the proof of Theorem 3.2, the

partially ordered sets (NI(A (S)), C ) are elementarily inequivalent, hence so are

the automorphism groups A ( S ) by Proposition 4.11. These remarks will be

further utilized in a later paper (see M. Droste, Normal subgroups and

elementary theories of lattice-ordered groups, to appear).

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complete embeddings of linear orderings and embeddings of lattice-ordered groups

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